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Digital differentiators enable the computation of the derivative of a continuous-time signal at discrete time instances, and they are used in many signal processing applications. This paper derives a unified filter order estimate for digital differentiators that are realized with linear-phase finite-length impulse response filters and designed in the minimax sense. The estimate is useful at the high-level system design when assessing the implementation complexity and it enables fewer designs when finding the minimal filter order required to satisfy a prescribed tolerable approximation error. The proposed unified estimate covers both wideband and lowpass differentiators of integer degrees up to ten. Furthermore, degree-individual filter order estimates are derived which improve and extend previous results. The performance of both the unified and degree-individual order estimates is evaluated through simulation examples and compared with previous estimates. [ABSTRACT FROM AUTHOR]

Inherent control delays and changeable grid operational conditions pose significant challenges to the stability of the Modular multilevel converter-based high voltage direct current (MMC-HVDC) system, inducing the risk of high-frequency resonances (HFR). This paper proposes a passivity enhancement based on the second-order digital differentiator (DD) and lead-lag compensator, ensuring reliable and robust HFR suppression performance even under the resonance frequency drift and fault earthing scenes. First, the passivity-based stability analysis is provided based on the system output impedance model, elucidating the HFR mechanisms and frequency drift problems. Second, the proposed passivity enhancement scheme's control structure and impedance reshaping principle are analyzed to verify resonance suppression ability. Third, the control parameters tuning process and direct realization of differentiator are detailed, reinforcing the stable margin and harmonic rejection ability. Finally, several case studies and simulation results are provided to validate the effectiveness of the proposed passivity-based HFR suppression scheme. • The proposed scheme integrates a second-order digital differentiator plus a lead-lag compensator. • Superior passivity and harmonic rejection ability are ensured, with frequency drift risk-solving ability. • Parameters tuning and differentiator discretization realization are elaborated. • Large phase lag and noise amplification problems are solved by impedance reshaping. [ABSTRACT FROM AUTHOR]

Design of full-band infinite impulse response (IIR) digital differentiator (DD) and digital integrator (DI) based on the Lk-norm and the min-max optimality criteria have been demonstrated in literature. However, the designed IIR DDs and DIs using these criteria do not have linear phase or have unsatisfactory magnitude response. In this paper, a novel approach to design, optimize and improve the magnitude response and the linearity of the phase response of IIR DD and DI using the cuckoo search (CS) optimization algorithm is proposed. The proposed approach is based on a varied Lk-norm where $k$ linearly increases with the number of the frequency samples between the values 1 and 2. The design and optimization of DDs and DIs are carried out using the CS optimization method due to its simplicity, efficiency, and robustness in solving general multidimensional optimization problems. The performance of the designed DDs and DIs using the proposed approach is compared with designs based on different criteria and algorithms available in literature. The comparison shows that the designed DDs and DIs based on the proposed approach have better phase responses and better or at least comparable magnitude responses compared to those of DDs and DIs designed using other methods in the literature.

In this paper, a novel design of third and fifth order differentiator based on lattice wave digital filter (LWDF), established on optimizing L_1-error approximation function using cuckoo search algorithm (CSA) is proposed. We present a novel realization of minimum multiplier differentiator using LWD structure leading to requirement of optimizing only N coefficients for Nth order differentiator. The gamma coefficients of lattice wave digital differentiator (LWDD) are computed by minimizing the L_1-norm fitness function leading to a flat response. The superiority of the proposed LWDD is evident by comparing it with other differentiators mentioned in the literature. The magnitude response of the designed LWDD is found to be of high accuracy with flat response in a wide frequency range. The simulation and statistical results validates that the designed minimum multiplier LWDD circumvents the existing one in terms of minimum absolute magnitude error, mean relative error (dB) and efficient structural realization, thereby making the proposed LWDD a promising approach to digital differentiator design.

A fractional-order digital differentiator is employed for the calculation of a time-derivative of the applied signal. In the recent few decades, this particular concept of a fractional derivative has been gaining a lot of attention in various applications concerning engineering, technology and science that includes image processing along with automatic control. Once there has been an effective use for this continuous-time Fractional-Order Differentiator (FOD), the trend in its research is primarily toward using a discrete-time fractional differentiator. All these conventional techniques tend to make use of a unimodal function for approximating an ideal FOD. For these techniques, there is a minimization of the fitness function that is accomplished by the algorithms which are based on the gradient. The fractional-order circuits along with their systems include an emerging area that has a high level of potential in aspects such as the biomedical instrumentation, control or signal processing. A digital differentiator is a tool that is extremely helpful in the determination and estimation of time derivatives of any given signal. Irrespective of the actual type of filter chosen (the Finite Impulse Response (FIR) or the Infinite-Length Impulse Response (IIR)), it is critical to bring down the complexity of computation needed for the implementation of the filter for a certain bandwidth and error of approximation. A metaheuristic algorithm normally has some advantages in the solving of problems which are Non-Deterministic Polynomial (NP)-hard. The Shuffled Frog Leaping Algorithm (SFLA) has been a new heuristic algorithm proposed in this work for the determination of optimal coefficients of the problem of FIR-FOD. A design for fractional-order-based digital differentiator is not a very important topic in research and signal processing. [ABSTRACT FROM AUTHOR]

Heart rate variability (HRV) analysis is considered as a preliminary diagnosis method to check the cardiac health of the human heart. The reliability of the HRV analysis system solely depends on the accuracy of the QRS complex detector. Hence, in this paper, an optimally designed digital differentiator (DD) for precise detection of QRS complex is proposed. The proposed DD is designed by using an efficient evolutionary optimization technique called gases Brownian motion optimization (GBMO) algorithm and is used in the preprocessing stage of the QRS detector. In GBMO algorithm, a balanced trade-off is maintained between both the exploration and the exploitation phases to find the global optimum solution. The electrocardiogram signal is preprocessed by using the proposed DD to generate the feature signals corresponding to the R-peaks only. The detection technique utilizes the principle of Hilbert transform and zeroes crossing detection. The proposed approach is verified against all the first channel records of MIT/BIH arrhythmia database by considering the standard QRS detection performance metrics and produces a sensitivity (Se) of 99.92%, positive predictivity (+P) of 99.92%, detection error rate (DER) of 0.1562%, QRS detection rate of 99.92%, accuracy (Acc) of 99.84%, and F score of 0.9992%. With respect to the standard performance metrics, the proposed QRS detector outperforms all the recently reported QRS detection techniques. [ABSTRACT FROM AUTHOR]

In this paper, a novel design of third and fifth order differentiator based on lattice wave digital filter (LWDF), established on optimizing L1-error approximation function using cuckoo search algorithm (CSA) is proposed. We present a novel realization of minimum multiplier differentiator using LWD structure leading to requirement of optimizing only N coefficients for N th order differentiator. The γ coefficients of lattice wave digital differentiator (LWDD) are computed by minimizing the L1 -norm fitness function leading to a flat response. The superiority of the proposed LWDD is evident by comparing it with other differentiators mentioned in the literature. The magnitude response of the designed LWDD is found to be of high accuracy with flat response in a wide frequency range. The simulation and statistical results validates that the designed minimum multiplier LWDD circumvents the existing one in terms of minimum absolute magnitude error, mean relative error (dB) and efficient structural realization, thereby making the proposed LWDD a promising approach to digital differentiator design. [ABSTRACT FROM AUTHOR]

New designs of recursive digital differentiators are obtained by optimizing a general fourth-order recursive digital filter over different Nyquist bands. In addition, another design of recursive digital differentiator is also obtained by optimizing the specified pole-zero locations of existing recursive digital differentiator of second-order system. Further, new designs of recursive digital integrators are obtained by inverting the transfer functions of designed recursive digital differentiators with suitable modifications. Thereafter, the zero-reflection approach is discussed and then applied to improve the phase responses of designed recursive digital differentiators and integrators. The beauty of finally obtained recursive digital differentiators and integrators is that they have nearly linear phase responses over wideband and also provide the choice of suitable recursive digital differentiator and integrator according to the importance of accuracy, bandwidth and the system simplicity.

In this paper, design of linear phase second order recursive digital integrators and differentiators is discussed. New second order integrators have been designed by using Genetic Algorithm (GA) optimization method. Thereafter, by modifying the transfer function of these integrators appropriately, new digital differentiators have been obtained. The proposed digital integrators and differentiators accurately approximate the ideal ones and have linear phase response over almost entire Nyquist frequency range. The proposed operators also outperform the existing operators in terms of both magnitude and phase response.

This paper presents a novel metaheuristic algorithm called Interior Search Algorithm (ISA), which is applied for digital differentiator design problem. ISA is based on the principles of aesthetic techniques commonly used in interior design and decoration. ISA has a very quick convergence rate and only one control parameter. The approach presented here has alleviated from the problems of premature convergence, stagnation and revisiting of the same solution over and over again, which is common in other optimization techniques. Statistical and simulation results have been compared with already existing differentiator design methods such as segment rule, genetic algorithm (GA), simulated annealing (SA), pole zero method (PZ) and particle swarm optimization (PSO). The results affirm that the proposed method outperforms its counterparts in terms of absolute magnitude error and phase error. [ABSTRACT FROM AUTHOR]

Recently, the Taylor–Fourier transform (TFT) was proposed to analyze the spectrum of signals with oscillating harmonics. The coefficients of this linear transformation were obtained through the calculation of the pseudoinverse matrix, which provides the classical solution to the normal equations of the least-squares (LS) approximation. This paper presents a filtering design technique that obtains the coefficients of the filters at each harmonic by imposing the maximally flat conditions to the polynomials defining their frequency responses. This condition can be used to solve the LS problem at each particular harmonic frequency, without the need of obtaining the whole set, as in the classical pseudoinverse solution. In addition, the filter passband central frequency can follow the fluctuations of the fundamental frequency. Besides, the method offers a reduction of the computational burden of the pseudoinverse solution. An implementation of the proposed estimator as an adaptive algorithm using its own instantaneous frequency estimate to relocate its bands is shown, and several tests are used to compare its performance with that of the ordinary TFT. [ABSTRACT FROM AUTHOR]

New IIR digital differintegrators (differentiator and integrator) with very minor absolute relative errors are presented in this paper. The digital integrator is designed by interpolating some of the existing integrators. The optimum value of the interpolation ratio is obtained through linear programming optimization. Subsequently, by modifying the transfer function of the proposed integrator appropriately, new digital differentiator is obtained. Simulation results demonstrate that the proposed differintegrator are a more accurate approximation of ideal ones, than the existing differintegrators. Furthermore, the proposed differentiator has been tested in an image processing application. Edges characterize boundaries and are, therefore, a problem of fundamental importance in image processing. For comparison purpose Prewitt, Sobel, Roberts, Canny, Laplacian of Gaussian (LOG), Zerocross operators were used and their results are displayed. The results of edge detection by some of the existing differentiators are also provided. The simulation results have shown the superiority of the proposed approach over existing ones. [ABSTRACT FROM AUTHOR]

New designs of recursive digital differentiators are obtained by optimizing a general fourth-order recursive digital filter over different Nyquist bands. In addition, another design of recursive digital differentiators is also obtained by optimizing the specified pole-zero locations of existing recursive digital differentiator of second-order system. Further, new designs of recursive digital integrators are obtained by inverting the transfer functions of designed recursive digital differentiators with suitable modifications. Thereafter, the zero-reflection approach is discussed and then applied to improve the phase responses of the designed recursive digital differentiators and integrators. The beauty of finally obtained recursive digital differentiators and integrators is that they have nearly linear phase responses over wideband and also provide the choice of suitable recursive digital differentiator and integrator according to the importance of accuracy, bandwidth and the system simplicity. [ABSTRACT FROM AUTHOR]

In this paper, design of linear phase second order recursive digital integrators and differentiators is discussed. New second order digital integrators have been designed by using Genetic Algorithm (GA) optimization method. Thereafter, by modifying the transfer function of these integrators appropriately, new digital differentiators have been obtained. The proposed digital integrators and differentiators accurately approximate the ideal ones and have linear phase response over almost entire Nyquist frequency range. The proposed operators also outperform the existing operators in terms of both magnitude and phase response. [ABSTRACT FROM AUTHOR]

This brief proposes a design method for a digital fractional order Savitzky-Golay differentiator (DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order for estimating the fractional order derivative of the contaminated signal. The proposed method calculates the moving window's weights using the polynomial least-squares method and the Riemann-Liouville fractional order derivative definition, and then computes the fractional order derivative of the given signal using the convolution between the weights and the signal, instead of the complex mathematical deduction. Hence, the computation time is greatly improved. Frequency-domain analysis reveals that the proposed differentiator is essentially a fractional order low-pass filter. Experiments demonstrate that the proposed DFOSGD can accurately estimate the fractional order derivatives of both noise-free signal and contaminated signal. [ABSTRACT FROM PUBLISHER]

In this correspondence, we propose a fractional bilinear transform used in analog-to-digital (A/D) conversion. This transfer function is derived by means of fractional delay filters. According to this transformation, first-order differentiator can be easily designed. Next, some design examples are illustrated to show the linear frequency mapping property when doing A/D conversion. Finally, the designed tow-order digitat differentiator is compared with the existing differentiators. [ABSTRACT FROM AUTHOR]

This brief presents a general theory of the Newton-Cotes digital integrators which is derived by applying the z-transform technique to the closed-form Newton-Cotes integration formula. Based on this developed theory, a new wideband third-order trapezoidal digital integrator is found to be a class of trapezoidal digital integrators. The novel wideband third-order trapezoidal integrator accurately approximates the ideal integrator over the whole Nyquist frequency range and compares favourably with existing integrators. Based on the designed wideband third-order trapezoidal integrator, a new wideband digital differentiator is designed, which approximates the ideal differentiator reasonably well over the whole Nyquist frequency range and compares favourably with existing differentiators. The low orders and high accuracies of the novel wideband trapezoidal integrator and the new wideband differentiator make them attractive for real-time applications. [ABSTRACT FROM AUTHOR]